A considerable proportion of the West Antarctic Ice Sheet sits on bedrock deep below sea level (an average depth of 1 km below sea level), forming a so-called marine ice sheet. The ice sheet thus lies partially submerged in the ocean, with the buoyancy of the ocean threatening to unpeel the base of the ice sheet from the bedrock. Since the bedrock slopes upwards in the direction of flow (an effect of the isostatic depression of the Antarctic continent by the ice sheet), the process of this unpeeling—or grounding-line retreat—can be exacerbated by a positive feedback mechanism created by the increasing buoyancy force as the grounding line retreats, a phenomenon known as the “Marine Ice Sheet Instability”. The collapse of regions of the ice sheet owing to this instability has the potential to add metres to sea level over the course of the next few centuries, and is a key unknown in efforts to predict future contributions to sea-level rise. This talk will review theoretical analysis of this instability and the associated extensional viscous thin-film theory, and demonstrate the dynamics through a combination of asymptotic and numerical studies, and laboratory experiments. The talk will continue by discussing an effect of the back force (buttressing) exerted by the floating sections of the ice sheet at its margins. It is shown that, subject to critical conditions, this effect will suppress instability.